Does the shape of the curve, that is produced in space by a mass, have a name ?

TheDinosaur

I have been reading an article about a theory of gravity that involves space being curved or warped.
An example was given where a heavy weight, representing the sun, is placed in the centre of a trampoline, and a golf ball, representing the earth, is caused to circle the centre of the trampoline due to the depression caused by the heavy weight.
I have two questions I would like to ask.
1) What is the name given to the shape of the curve that is produced in
space by a mass ?
I remember studying Conic Sections when I was at school and just
as you can derive many different parabolic curves from one cone, they
are all called a parabola.
Equally, although different masses will cause different curvatures of
space, I presume that all these curves will come under just one name.
Just as I am able to understand the relationship between a parabola and
a cone, my wish is to look up the definition of this name in order to
understand what the underlying relationship of the curve is.
( I understand that space can be looked upon as being a three
dimensional surface curved through four dimensional space. This is
analogous to the two dimensional surface of the trampoline being curved
through three dimensional space. )

2) At what point, and how, does the trampoline analogy break down ?
My understanding, from the article, is that the direction that the
earth takes in space, is completely controlled by the curvature of the
space itself.
I can certainly understand that the curvature will influence the
direction the Earth takes but not control it. From the trampoline example,
there are essentially two things controlling the direction of the golf ball.
One is the curvature of the trampoline and the other ( ignoring friction )
is the force of gravity acting downwards on the ball. However, as far as I
can see, there is no equivalent in the Sun-Earth system to gravity acting
downwards on the golf ball.
If we imagine our three dimensional universe to be the three
dimensional surface of a four dimensional sphere then, to continue the
analogy with the trampoline, the force of gravity acting on the golf ball
would be equivalent to a force acting on matter towards the centre of
the four dimensional sphere. Thus, this force pulls the Earth down into
the depression caused by the Sun and the Sun is also pulled on by this
force.
The implication, from the article, is that the direction that the
earth takes in space, is completely controlled by the curvature of the
space that the earth travels through. How can this be ?
( Perhaps I should note here that I may be having difficulty with
the fact that the depression in the surface of the trampoline is negatively
curved. Certainly, if we were dealing with a positively curved surface, ie.
a sphere around the sun that the Earth travelled on, then there would be
no doubt that the Earth would be forced to orbit the Sun on a geodesic.)
I apologise about being so long winded with my questions. However, my feeling is that, if I am able to articulate where my thinking lies, then you will more easily be able to comment on my misunderstandings and point me to where I can do further reading.
I hope I have chosen the correct forum to post to and I hope my questions aren’t too basic for this forum.

mutant

"However, as far as I
can see, there is no equivalent in the Sun-Earth system to gravity acting
downwards on the golf ball."

This is where the analogy breaks down: at the first step.
A better analogy can be found in Taylor-Wheeler's Space-time Physics, two men walkig along meridians of Earth. Do you have this?
m

A.T.

The trampoline is an analogy for curved space. But you need curved spacetime to understand how gravitation works. Try this:
http://fy.chalmers.se/~rico/Theses/tesx.pdf
http://www.adamtoons.de/physics/gravitation.swf

TheDinosaur

Thank you mutant and A.T.for your replies.
No, I have never heard of Taylor-Wheeler's Space-time Physics but a Google search shows it to be well recommended
Gravity Illustrated Spacetime Edition also seems to be ideal if I can cope with a Doctoral Thesis.

You have both let me know I should be reading about spacetime and I am very greatful to know this.

By the way, A.T.’s link http://www.adamtoons.de/physics/gravitation.swf just brings up a Black ( and I do mean Black and not blank ) page. It does this in both Mozilla and Internet Explorer. I am very interested to see what the link contains. I went to http://www.adamtoons.de/ and found it contains the link http://www.adamtoons.de/physics/gravitation.swf so I presume I don’t have a plugin. I was wondering if A.T. could please suggest what I can do to view the page.

A.T.

You don't need the math of the later chapters to understand the idea. Just read chapter 2 und try to understand the pictures.

You need the flash plugin:
http://www.adobe.com/products/flashplayer/
Some browsers are also blocking active content depending on security settings.

Wonderballs

I thought gravity was the curve produced in spacetime...

pervect

Yes, gravity requires a curved space-time, not just "curved space". In fact, with the usual slicing of space-time into space+time, the curvature in the space part is present but negligible at ordinary velocities - it is the non-spatial curvature that is important.

(This result can vary with different ways of "slicing" space-time, but the usual Schwarzschild coordinate time acts in this manner).